ARMA Model - stock (Reliance)
# ARMA (0,0)
arma00 <- arima(stock, order = c(0,0,0)) # log likelihood = -1437.64, aic = 2879.28
summary(arma00)
##
## Call:
## arima(x = stock, order = c(0, 0, 0))
##
## Coefficients:
## intercept
## 0.0053
## s.e. 0.0515
##
## sigma^2 estimated as 2.121: log likelihood = -1437.64, aic = 2879.28
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -7.48465e-17 1.456253 1.061418 100.1662 100.6747 0.688961
## ACF1
## Training set -0.0161454
# ARMA(1,0)
arma10 <- arima(stock, order = c(1,0,0)) # log likelihood = -1437.53, aic = 2881.07
summary(arma10)
##
## Call:
## arima(x = stock, order = c(1, 0, 0))
##
## Coefficients:
## ar1 intercept
## -0.0161 0.0053
## s.e. 0.0353 0.0506
##
## sigma^2 estimated as 2.12: log likelihood = -1437.53, aic = 2881.07
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 2.543475e-05 1.456062 1.061917 102.6262 102.9787 0.6892852
## ACF1
## Training set 0.0001990777
# ARMA(0,1)
arma01 <- arima(stock, order = c(0,0,1)) # log likelihood = -1437.54, aic = 2881.07
summary(arma01)
##
## Call:
## arima(x = stock, order = c(0, 0, 1))
##
## Coefficients:
## ma1 intercept
## -0.0158 0.0053
## s.e. 0.0350 0.0506
##
## sigma^2 estimated as 2.12: log likelihood = -1437.54, aic = 2881.07
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 6.919107e-06 1.456067 1.06191 102.5167 102.8533 0.6892801
## ACF1
## Training set -0.0001635539
# ARMA(1,1)
arma11 <- arima(stock, order = c(1,0,1)) # log likelihood = -1436.76, aic = 2881.53
summary(arma11)
##
## Call:
## arima(x = stock, order = c(1, 0, 1))
##
## Coefficients:
## ar1 ma1 intercept
## 0.8829 -0.9050 0.0052
## s.e. 0.1504 0.1364 0.0419
##
## sigma^2 estimated as 2.116: log likelihood = -1436.76, aic = 2881.53
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.0008050277 1.454651 1.06254 112.3929 118.5012 0.6896893
## ACF1
## Training set 0.003824042
# ARMA(2,0)
arma20 <- arima(stock, order = c(2,0,0)) # log likelihood = -1437.47, aic = 2882.94
summary(arma20)
##
## Call:
## arima(x = stock, order = c(2, 0, 0))
##
## Coefficients:
## ar1 ar2 intercept
## -0.0159 0.0126 0.0053
## s.e. 0.0353 0.0354 0.0513
##
## sigma^2 estimated as 2.12: log likelihood = -1437.47, aic = 2882.94
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -3.417605e-05 1.455946 1.061706 105.67 106.8302 0.6891476
## ACF1
## Training set 0.001085547
# ARMA(0,2)
arma02 <- arima(stock, order = c(0,0,2)) # log likelihood = -1437.5, aic = 2882.99
summary(arma02)
##
## Call:
## arima(x = stock, order = c(0, 0, 2))
##
## Coefficients:
## ma1 ma2 intercept
## -0.0141 0.0103 0.0053
## s.e. 0.0358 0.0357 0.0512
##
## sigma^2 estimated as 2.12: log likelihood = -1437.5, aic = 2882.99
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -1.621463e-05 1.455991 1.061668 104.6583 105.3018 0.6891234
## ACF1
## Training set -0.0009257076
# ARMA(2,2)
arma22 <- arima(stock, order = c(2,0,2)) # log likelihood = -1432.85, aic = 2877.71
summary(arma22)
##
## Call:
## arima(x = stock, order = c(2, 0, 2))
##
## Coefficients:
## ar1 ar2 ma1 ma2 intercept
## -0.9874 -0.9656 1.0103 0.9997 0.0059
## s.e. 0.0145 0.0107 0.0116 0.0195 0.0520
##
## sigma^2 estimated as 2.085: log likelihood = -1432.85, aic = 2877.71
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -0.0005299047 1.443957 1.057919 96.60171 151.0894 0.68669
## ACF1
## Training set -0.03007662
#ARMA(3,0)
arma30 <- arima(stock, order = c(3,0,0)) # log likelihood = -1434.36, aic = 2878.72
summary(arma30)
##
## Call:
## arima(x = stock, order = c(3, 0, 0))
##
## Coefficients:
## ar1 ar2 ar3 intercept
## -0.0147 0.0113 -0.0881 0.0052
## s.e. 0.0352 0.0352 0.0352 0.0470
##
## sigma^2 estimated as 2.103: log likelihood = -1434.36, aic = 2878.72
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.0001246886 1.450282 1.063358 148.4398 163.3296 0.6902199
## ACF1
## Training set 0.0001496099
#ARMA(0,3)
arma03 <- arima(stock, order = c(0,0,3)) # log likelihood = -1434.68, aic = 2879.35
summary(arma03)
##
## Call:
## arima(x = stock, order = c(0, 0, 3))
##
## Coefficients:
## ma1 ma2 ma3 intercept
## -0.0148 0.0089 -0.0801 0.0052
## s.e. 0.0353 0.0353 0.0337 0.0469
##
## sigma^2 estimated as 2.105: log likelihood = -1434.68, aic = 2879.35
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 9.748187e-05 1.450859 1.063395 142.8417 154.8663 0.690244
## ACF1
## Training set -0.0002213725
# ARMA(3,3)
arma33 <- arima(stock, order = c(3,0,3)) # log likelihood = -1431.77, aic = 2879.54
summary(arma33)
##
## Call:
## arima(x = stock, order = c(3, 0, 3))
##
## Coefficients:
## Warning in sqrt(diag(x$var.coef)): NaNs produced
## ar1 ar2 ar3 ma1 ma2 ma3 intercept
## -0.1242 -0.1148 0.831 0.1219 0.1026 -0.888 0.0050
## s.e. NaN NaN NaN NaN NaN 0.046 0.0421
##
## sigma^2 estimated as 2.079: log likelihood = -1431.77, aic = 2879.54
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001017894 1.441909 1.058898 109.6531 158.152 0.6873255
## ACF1
## Training set -0.007072237
Best Model - Stock (reliance)
cat("Best Model is ----> ** arma(2,2) ** ")
## Best Model is ----> ** arma(2,2) **
# Among all the model,
#(2,2) has lowest AIC with aic(2,2) = 2877.71,
# Also, coefficient ar1,ar2 and ma1, ma2 is statstically significant
arma_residuals <- residuals(arma22)
#Heteroskedasticity
ArchTest(arma_residuals, lags = 10)
##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: arma_residuals
## Chi-squared = 21.061, df = 10, p-value = 0.02067
# Presence of ARCH effects => consider GARCH models
# White noise
Box.test(arma_residuals, lag = 10, type = "Ljung-Box") # p-value = 0.3489
##
## Box-Ljung test
##
## data: arma_residuals
## X-squared = 11.111, df = 10, p-value = 0.3489
# Residuals are not autocorrelated and behave like white noise
# Normality test
shapiro.test(arma_residuals)
##
## Shapiro-Wilk normality test
##
## data: arma_residuals
## W = 0.96907, p-value = 5.447e-12
# Residuals are not normally distributed (presence of tails, skewness, or kurtosis)
# ACF
asset_name = "(Residuals - Reliance) - "
acf(arma_residuals, main = paste(asset_name, "ACF"))
ARMA Model - Index (Nifty 500)
# ARMA (0,0)
Arma00 <- arima(index, order = c(0,0,0)) # log likelihood = -1047.84, aic = 2099.67
summary(Arma00)
##
## Call:
## arima(x = index, order = c(0, 0, 0))
##
## Coefficients:
## intercept
## 0.0373
## s.e. 0.0316
##
## sigma^2 estimated as 0.8013: log likelihood = -1047.84, aic = 2099.67
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 6.531232e-15 0.8951382 0.6526778 -Inf Inf 0.7330358 -0.01728411
# ARMA(1,0)
Arma10 <- arima(index, order = c(1,0,0)) # log likelihood = -1047.72, aic = 2101.44
summary(Arma10)
##
## Call:
## arima(x = index, order = c(1, 0, 0))
##
## Coefficients:
## ar1 intercept
## -0.0173 0.0373
## s.e. 0.0353 0.0311
##
## sigma^2 estimated as 0.801: log likelihood = -1047.72, aic = 2101.44
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 1.853828e-05 0.8950043 0.6533607 -Inf Inf 0.7338027 0.0001328109
# ARMA(0,1)
Arma01 <- arima(index, order = c(0,0,1)) # log likelihood = -1047.72, aic = 2101.44
summary(Arma01)
##
## Call:
## arima(x = index, order = c(0, 0, 1))
##
## Coefficients:
## ma1 intercept
## -0.0170 0.0373
## s.e. 0.0351 0.0311
##
## sigma^2 estimated as 0.801: log likelihood = -1047.72, aic = 2101.44
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 8.678214e-06 0.8950063 0.653351 -Inf Inf 0.7337919 -0.0001291844
# ARMA(1,1)
Arma11 <- arima(index, order = c(1,0,1)) # log likelihood = -1047.25, aic = 2102.51
## Warning in arima(index, order = c(1, 0, 1)): possible convergence problem:
## optim gave code = 1
summary(Arma11)
##
## Call:
## arima(x = index, order = c(1, 0, 1))
##
## Coefficients:
## ar1 ma1 intercept
## -0.7617 0.7367 0.0360
## s.e. 0.2842 0.2914 0.0312
##
## sigma^2 estimated as 0.8001: log likelihood = -1047.25, aic = 2102.51
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 0.001317732 0.8944832 0.6530551 -Inf Inf 0.7334595 0.009083419
# ARMA(2,0)
Arma20 <- arima(index, order = c(2,0,0)) # log likelihood = -1047.69, aic = 2103.39
summary(Arma20)
##
## Call:
## arima(x = index, order = c(2, 0, 0))
##
## Coefficients:
## ar1 ar2 intercept
## -0.0171 0.0078 0.0373
## s.e. 0.0353 0.0353 0.0313
##
## sigma^2 estimated as 0.801: log likelihood = -1047.69, aic = 2103.39
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 4.068677e-07 0.8949774 0.6533283 -Inf Inf 0.7337664 0.0001164638
# ARMA(0,2)
Arma02 <- arima(index, order = c(0,0,2)) # log likelihood = -1047.7, aic = 2103.4
summary(Arma02)
##
## Call:
## arima(x = index, order = c(0, 0, 2))
##
## Coefficients:
## ma1 ma2 intercept
## -0.0169 0.0069 0.0373
## s.e. 0.0353 0.0339 0.0313
##
## sigma^2 estimated as 0.801: log likelihood = -1047.7, aic = 2103.4
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set -2.352259e-06 0.8949829 0.6533262 -Inf Inf 0.733764 -0.0001081153
# ARMA(2,2)
Arma22 <- arima(index, order = c(2,0,2)) # log likelihood = -1046.07, aic = 2104.14
## Warning in arima(index, order = c(2, 0, 2)): possible convergence problem:
## optim gave code = 1
summary(Arma22)
##
## Call:
## arima(x = index, order = c(2, 0, 2))
##
## Coefficients:
## Warning in sqrt(diag(x$var.coef)): NaNs produced
## ar1 ar2 ma1 ma2 intercept
## -1.7171 -0.8876 1.7185 0.9016 0.0331
## s.e. 0.0337 NaN NaN NaN 0.0317
##
## sigma^2 estimated as 0.7977: log likelihood = -1046.07, aic = 2104.14
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 0.004262089 0.8931376 0.652483 -Inf Inf 0.7328169 -0.01438681
Best ARMA Model - Index (Nifty 50)
cat("Best Model is ----> ** Arma(0,0) ** ")
## Best Model is ----> ** Arma(0,0) **
# arma(0,0) has the lowest AIC = 2099.67
Arma_residuals <- residuals(Arma00)
#Heteroskedasticity
ArchTest(Arma_residuals, lags = 10)
##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: Arma_residuals
## Chi-squared = 94.778, df = 10, p-value = 6.015e-16
# Presence of ARCH effects => consider GARCH models
# White noise
Box.test(Arma_residuals, lag = 10, type = "Ljung-Box") # p-value = 0.859
##
## Box-Ljung test
##
## data: Arma_residuals
## X-squared = 4.0099, df = 10, p-value = 0.9469
# Residuals are not autocorrelated and behave like white noise
# Normality test
shapiro.test(Arma_residuals)
##
## Shapiro-Wilk normality test
##
## data: Arma_residuals
## W = 0.95772, p-value = 1.996e-14
# Residuals are not normally distributed (presence of tails, skewness, or kurtosis)
# ACF
asset_name = "(Residuals - Nifty 50) - "
acf(Arma_residuals, main = paste(asset_name, "ACF"))
Garch Model - Stock (Reliance) With arma(2,2)
# Mean model spec (ARMA(2,2) with mean)
mean_ <- list(armaOrder = c(2,2), include.mean = TRUE)
# GARCH(1,1)
spec_garch <- ugarchspec(
variance.model = list(model = "sGARCH", garchOrder = c(1,1)),
mean.model = mean_,
distribution.model = "std"
)
fit_garch <- ugarchfit(spec_garch, data = stock)
show(fit_garch)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.004495 0.044226 0.10163 0.919052
## ar1 -0.253036 0.003378 -74.90400 0.000000
## ar2 -0.985059 0.005184 -190.01784 0.000000
## ma1 0.262902 0.000974 269.89101 0.000000
## ma2 1.003423 0.000166 6054.73812 0.000000
## omega 0.023862 0.012568 1.89867 0.057608
## alpha1 0.024658 0.008056 3.06081 0.002207
## beta1 0.963140 0.009927 97.02372 0.000000
## shape 5.177255 0.880597 5.87925 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.004495 0.042701 0.10526 0.916171
## ar1 -0.253036 0.003616 -69.97141 0.000000
## ar2 -0.985059 0.004646 -212.04243 0.000000
## ma1 0.262902 0.000881 298.26753 0.000000
## ma2 1.003423 0.000143 7032.97975 0.000000
## omega 0.023862 0.010891 2.19105 0.028448
## alpha1 0.024658 0.006793 3.62992 0.000284
## beta1 0.963140 0.006332 152.11617 0.000000
## shape 5.177255 0.900127 5.75170 0.000000
##
## LogLikelihood : -1378.746
##
## Information Criteria
## ------------------------------------
##
## Akaike 3.4650
## Bayes 3.5177
## Shibata 3.4648
## Hannan-Quinn 3.4853
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.2853 0.59324
## Lag[2*(p+q)+(p+q)-1][11] 7.3386 0.01759
## Lag[4*(p+q)+(p+q)-1][19] 9.3805 0.57354
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 10.21 0.001395
## Lag[2*(p+q)+(p+q)-1][5] 12.58 0.001959
## Lag[4*(p+q)+(p+q)-1][9] 13.68 0.007392
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 3.483 0.500 2.000 0.0620
## ARCH Lag[5] 4.133 1.440 1.667 0.1623
## ARCH Lag[7] 4.292 2.315 1.543 0.3064
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.6705
## Individual Statistics:
## mu 0.07643
## ar1 0.23622
## ar2 0.07354
## ma1 0.04569
## ma2 0.16970
## omega 0.20743
## alpha1 0.50854
## beta1 0.36649
## shape 0.49336
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.1 2.32 2.82
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.1447 0.25268
## Negative Sign Bias 0.6743 0.50033
## Positive Sign Bias 1.9046 0.05719 *
## Joint Effect 4.1443 0.24630
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 23.04 0.2354
## 2 30 25.93 0.6293
## 3 40 34.71 0.6661
## 4 50 49.50 0.4532
##
##
## Elapsed time : 0.1988499
# GJR-GARCH(1,1)
spec_gjr <- ugarchspec(
variance.model = list(model = "gjrGARCH", garchOrder = c(1,1)),
mean.model = mean_,
distribution.model = "std"
)
fit_gjr <- ugarchfit(spec_gjr, data = stock)
show(fit_gjr)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.003460 0.044633 0.077513 0.93822
## ar1 -1.594729 0.029236 -54.547471 0.00000
## ar2 -0.703995 0.041682 -16.889617 0.00000
## ma1 1.603377 0.025161 63.725599 0.00000
## ma2 0.730126 0.034067 21.432137 0.00000
## omega 0.020210 0.024915 0.811166 0.41727
## alpha1 0.019816 0.013148 1.507124 0.13178
## beta1 0.967193 0.023854 40.545844 0.00000
## gamma1 0.005345 0.019440 0.274933 0.78337
## shape 5.204163 0.917178 5.674104 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.003460 0.043470 0.079587 0.936566
## ar1 -1.594729 0.042492 -37.530261 0.000000
## ar2 -0.703995 0.022920 -30.714898 0.000000
## ma1 1.603377 0.028860 55.556460 0.000000
## ma2 0.730126 0.004670 156.339142 0.000000
## omega 0.020210 0.038610 0.523452 0.600660
## alpha1 0.019816 0.017613 1.125111 0.260542
## beta1 0.967193 0.036006 26.862314 0.000000
## gamma1 0.005345 0.021785 0.245342 0.806191
## shape 5.204163 1.043086 4.989199 0.000001
##
## LogLikelihood : -1382.629
##
## Information Criteria
## ------------------------------------
##
## Akaike 3.4772
## Bayes 3.5357
## Shibata 3.4769
## Hannan-Quinn 3.4997
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.2121 0.6452
## Lag[2*(p+q)+(p+q)-1][11] 3.6668 1.0000
## Lag[4*(p+q)+(p+q)-1][19] 4.8133 0.9960
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 12.07 0.0005129
## Lag[2*(p+q)+(p+q)-1][5] 14.34 0.0006663
## Lag[4*(p+q)+(p+q)-1][9] 15.28 0.0030558
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 3.359 0.500 2.000 0.06684
## ARCH Lag[5] 3.832 1.440 1.667 0.18968
## ARCH Lag[7] 3.903 2.315 1.543 0.36080
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.5717
## Individual Statistics:
## mu 0.07847
## ar1 0.08516
## ar2 0.03335
## ma1 0.11008
## ma2 0.05237
## omega 0.19667
## alpha1 0.47640
## beta1 0.34799
## gamma1 0.51915
## shape 0.49625
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.29 2.54 3.05
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.920 0.055213 *
## Negative Sign Bias 1.078 0.281487
## Positive Sign Bias 2.599 0.009535 ***
## Joint Effect 7.916 0.047770 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 17.65 0.5458
## 2 30 24.21 0.7187
## 3 40 40.70 0.3956
## 4 50 35.64 0.9232
##
##
## Elapsed time : 0.2978539
# EGARCH(1,1)
spec_egarch <- ugarchspec(
variance.model = list(model = "eGARCH", garchOrder = c(1,1)),
mean.model = mean_,
distribution.model = "std"
)
fit_egarch <- ugarchfit(spec_egarch, data = stock)
show(fit_egarch)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(1,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.006099 0.043731 1.3946e-01 0.88909
## ar1 -1.120038 0.000067 -1.6727e+04 0.00000
## ar2 -1.032701 0.000066 -1.5647e+04 0.00000
## ma1 1.120038 0.000071 1.5758e+04 0.00000
## ma2 1.032701 0.000029 3.5270e+04 0.00000
## omega 0.008260 0.002723 3.0332e+00 0.00242
## alpha1 -0.011337 0.017936 -6.3210e-01 0.52732
## beta1 0.987248 0.000015 6.6383e+04 0.00000
## gamma1 0.071843 0.006673 1.0767e+01 0.00000
## shape 5.263153 0.930160 5.6583e+00 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.006099 0.168333 0.036230 0.97110
## ar1 -1.120038 0.017618 -63.575222 0.00000
## ar2 -1.032701 0.011297 -91.411657 0.00000
## ma1 1.120038 0.017721 63.203603 0.00000
## ma2 1.032701 0.024142 42.776000 0.00000
## omega 0.008260 0.332964 0.024808 0.98021
## alpha1 -0.011337 0.473519 -0.023942 0.98090
## beta1 0.987248 0.006636 148.782564 0.00000
## gamma1 0.071843 1.044588 0.068777 0.94517
## shape 5.263153 159.262109 0.033047 0.97364
##
## LogLikelihood : -1382.217
##
## Information Criteria
## ------------------------------------
##
## Akaike 3.4762
## Bayes 3.5347
## Shibata 3.4759
## Hannan-Quinn 3.4987
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.07184 0.7886774
## Lag[2*(p+q)+(p+q)-1][11] 8.30877 0.0002726
## Lag[4*(p+q)+(p+q)-1][19] 10.44214 0.3966313
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 11.02 0.000901
## Lag[2*(p+q)+(p+q)-1][5] 13.34 0.001236
## Lag[4*(p+q)+(p+q)-1][9] 14.37 0.005059
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 3.626 0.500 2.000 0.05688
## ARCH Lag[5] 3.950 1.440 1.667 0.17847
## ARCH Lag[7] 4.092 2.315 1.543 0.33354
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 14.6898
## Individual Statistics:
## mu 0.08568
## ar1 0.29267
## ar2 0.18840
## ma1 0.29327
## ma2 0.21050
## omega 0.28051
## alpha1 0.22711
## beta1 0.48314
## gamma1 0.11503
## shape 0.41463
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.29 2.54 3.05
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.821 0.06898 *
## Negative Sign Bias 1.213 0.22557
## Positive Sign Bias 2.551 0.01092 **
## Joint Effect 8.000 0.04602 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 13.71 0.8005
## 2 30 20.54 0.8755
## 3 40 27.41 0.9181
## 4 50 47.25 0.5442
##
##
## Elapsed time : 0.569818
# IGARCH(1,1)
spec_igarch <- ugarchspec(
variance.model = list(model = "iGARCH", garchOrder = c(1,1)),
mean.model = mean_,
distribution.model = "std"
)
fit_igarch <- ugarchfit(spec_igarch, data = stock)
show(fit_igarch)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : iGARCH(1,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.008753 0.044402 0.19714 0.843717
## ar1 -1.596360 0.022595 -70.65127 0.000000
## ar2 -0.701083 0.039875 -17.58200 0.000000
## ma1 1.602299 0.020942 76.51027 0.000000
## ma2 0.724197 0.032233 22.46722 0.000000
## omega 0.001409 0.003792 0.37157 0.710214
## alpha1 0.018023 0.008387 2.14890 0.031642
## beta1 0.981977 NA NA NA
## shape 4.882789 0.793912 6.15029 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.008753 0.045533 0.19224 0.84755
## ar1 -1.596360 0.039262 -40.65885 0.00000
## ar2 -0.701083 0.025902 -27.06695 0.00000
## ma1 1.602299 0.025164 63.67345 0.00000
## ma2 0.724197 0.004700 154.09582 0.00000
## omega 0.001409 0.004971 0.28340 0.77687
## alpha1 0.018023 0.011260 1.60069 0.10945
## beta1 0.981977 NA NA NA
## shape 4.882789 0.893594 5.46421 0.00000
##
## LogLikelihood : -1383.796
##
## Information Criteria
## ------------------------------------
##
## Akaike 3.4751
## Bayes 3.5219
## Shibata 3.4749
## Hannan-Quinn 3.4931
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.302 0.5826
## Lag[2*(p+q)+(p+q)-1][11] 3.716 1.0000
## Lag[4*(p+q)+(p+q)-1][19] 4.880 0.9954
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 15.51 8.204e-05
## Lag[2*(p+q)+(p+q)-1][5] 18.66 4.516e-05
## Lag[4*(p+q)+(p+q)-1][9] 19.80 2.264e-04
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 5.052 0.500 2.000 0.02459
## ARCH Lag[5] 5.248 1.440 1.667 0.09003
## ARCH Lag[7] 5.277 2.315 1.543 0.19795
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.2827
## Individual Statistics:
## mu 0.08462
## ar1 0.08036
## ar2 0.02898
## ma1 0.10792
## ma2 0.04696
## omega 0.06879
## alpha1 0.19252
## shape 0.28142
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.89 2.11 2.59
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.043 0.041367 **
## Negative Sign Bias 1.129 0.259376
## Positive Sign Bias 2.823 0.004872 ***
## Joint Effect 9.245 0.026203 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 13.26 0.8251
## 2 30 23.98 0.7299
## 3 40 29.41 0.8671
## 4 50 37.51 0.8843
##
##
## Elapsed time : 0.09199214
# APARCH(1,1)
spec_aparch <- ugarchspec(
variance.model = list(model = "apARCH", garchOrder = c(1,1)),
mean.model = mean_,
distribution.model = "std"
)
fit_aparch <- ugarchfit(spec_aparch, data = stock)
show(fit_aparch)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : apARCH(1,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.003011 0.043884 6.8607e-02 0.94530
## ar1 -1.812423 0.003523 -5.1450e+02 0.00000
## ar2 -0.982901 0.002207 -4.4533e+02 0.00000
## ma1 1.824259 0.003624 5.0341e+02 0.00000
## ma2 0.996060 0.000125 7.9587e+03 0.00000
## omega 0.006784 0.005528 1.2272e+00 0.21975
## alpha1 0.000171 0.000186 9.2023e-01 0.35745
## beta1 0.991649 0.000953 1.0411e+03 0.00000
## gamma1 -0.999999 0.000745 -1.3427e+03 0.00000
## delta 3.500000 0.403200 8.6806e+00 0.00000
## shape 5.167394 1.013852 5.0968e+00 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.003011 NaN NaN NaN
## ar1 -1.812423 NaN NaN NaN
## ar2 -0.982901 NaN NaN NaN
## ma1 1.824259 NaN NaN NaN
## ma2 0.996060 NaN NaN NaN
## omega 0.006784 NaN NaN NaN
## alpha1 0.000171 NaN NaN NaN
## beta1 0.991649 NaN NaN NaN
## gamma1 -0.999999 NaN NaN NaN
## delta 3.500000 NaN NaN NaN
## shape 5.167394 NaN NaN NaN
##
## LogLikelihood : -1377.869
##
## Information Criteria
## ------------------------------------
##
## Akaike 3.4678
## Bayes 3.5322
## Shibata 3.4675
## Hannan-Quinn 3.4926
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.5644 0.45248
## Lag[2*(p+q)+(p+q)-1][11] 6.9034 0.07187
## Lag[4*(p+q)+(p+q)-1][19] 8.4309 0.73087
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 13.86 0.0001968
## Lag[2*(p+q)+(p+q)-1][5] 16.52 0.0001732
## Lag[4*(p+q)+(p+q)-1][9] 17.55 0.0008437
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 4.257 0.500 2.000 0.03908
## ARCH Lag[5] 4.489 1.440 1.667 0.13467
## ARCH Lag[7] 4.575 2.315 1.543 0.27110
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: NaN
## Individual Statistics:
## mu 0.12019
## ar1 0.03289
## ar2 0.04268
## ma1 0.01824
## ma2 0.02405
## omega 0.15889
## alpha1 0.11731
## beta1 0.14395
## gamma1 NaN
## delta 0.11294
## shape 0.18619
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.49 2.75 3.27
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.5874 0.11282
## Negative Sign Bias 0.6834 0.49456
## Positive Sign Bias 2.7563 0.00598 ***
## Joint Effect 8.1100 0.04379 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 9.262 0.9689
## 2 30 18.363 0.9365
## 3 40 26.316 0.9397
## 4 50 29.524 0.9875
##
##
## Elapsed time : 1.337209
# GARCH-in-Mean(1,1)
mean_garchm <- list(armaOrder = c(2,2), include.mean = TRUE, archm = TRUE)
spec_garchm <- ugarchspec(
variance.model = list(model = "sGARCH", garchOrder = c(1,1)),
mean.model = mean_garchm,
distribution.model = "std"
)
fit_garchm <- ugarchfit(spec_garchm, data = stock)
show(fit_garchm)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu -0.222794 0.643395 -0.34628 0.729133
## ar1 -1.592030 0.012759 -124.77310 0.000000
## ar2 -0.705297 0.036872 -19.12840 0.000000
## ma1 1.601938 0.016602 96.49258 0.000000
## ma2 0.732979 0.027833 26.33533 0.000000
## archm 0.164738 0.466371 0.35323 0.723914
## omega 0.036527 0.035451 1.03035 0.302848
## alpha1 0.030523 0.014744 2.07017 0.038437
## beta1 0.951465 0.028460 33.43154 0.000000
## shape 5.143152 0.891572 5.76863 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu -0.222794 1.697209 -0.13127 0.89556
## ar1 -1.592030 0.031501 -50.53950 0.00000
## ar2 -0.705297 0.022795 -30.94143 0.00000
## ma1 1.601938 0.017245 92.89417 0.00000
## ma2 0.732979 0.008565 85.58039 0.00000
## archm 0.164738 1.229109 0.13403 0.89338
## omega 0.036527 0.086418 0.42268 0.67253
## alpha1 0.030523 0.027020 1.12962 0.25864
## beta1 0.951465 0.066400 14.32933 0.00000
## shape 5.143152 0.994143 5.17345 0.00000
##
## LogLikelihood : -1382.597
##
## Information Criteria
## ------------------------------------
##
## Akaike 3.4771
## Bayes 3.5356
## Shibata 3.4768
## Hannan-Quinn 3.4996
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1256 0.7230
## Lag[2*(p+q)+(p+q)-1][11] 3.4331 1.0000
## Lag[4*(p+q)+(p+q)-1][19] 4.5309 0.9978
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 9.232 0.002378
## Lag[2*(p+q)+(p+q)-1][5] 11.124 0.004717
## Lag[4*(p+q)+(p+q)-1][9] 12.050 0.017667
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 2.435 0.500 2.000 0.1187
## ARCH Lag[5] 3.235 1.440 1.667 0.2575
## ARCH Lag[7] 3.354 2.315 1.543 0.4494
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.8254
## Individual Statistics:
## mu 0.10933
## ar1 0.09906
## ar2 0.04389
## ma1 0.12212
## ma2 0.06530
## archm 0.08599
## omega 0.26548
## alpha1 0.59439
## beta1 0.48509
## shape 0.55743
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.29 2.54 3.05
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.293 0.02214 **
## Negative Sign Bias 1.368 0.17172
## Positive Sign Bias 2.457 0.01422 **
## Joint Effect 8.138 0.04324 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 22.10 0.2795
## 2 30 20.39 0.8805
## 3 40 42.80 0.3115
## 4 50 47.75 0.5238
##
##
## Elapsed time : 0.3812749
# Compare Information Criteria
infocriteria(fit_garch)
##
## Akaike 3.465035
## Bayes 3.517685
## Shibata 3.464786
## Hannan-Quinn 3.485259
infocriteria(fit_gjr)
##
## Akaike 3.477226
## Bayes 3.535726
## Shibata 3.476919
## Hannan-Quinn 3.499698
infocriteria(fit_egarch)
##
## Akaike 3.476196
## Bayes 3.534697
## Shibata 3.475890
## Hannan-Quinn 3.498668
infocriteria(fit_igarch)
##
## Akaike 3.475146
## Bayes 3.521946
## Shibata 3.474949
## Hannan-Quinn 3.493123
infocriteria(fit_aparch)
##
## Akaike 3.467837
## Bayes 3.532187
## Shibata 3.467467
## Hannan-Quinn 3.492556
infocriteria(fit_garchm)
##
## Akaike 3.477146
## Bayes 3.535646
## Shibata 3.476839
## Hannan-Quinn 3.499618
Best Garch Model - Stock (Reliance)
# GARCH(1,1): Akaike (AIC): 3.465035, Bayes (BIC): 3.517685 -> Lowest AIC
# GJR-GARCH(1,1): Akaike (AIC): 3.477226, Bayes (BIC): 3.535726
# EGARCH(1,1): Akaike (AIC): 3.476196, Bayes (BIC): 3.534697
# IGARCH(1,1): Akaike (AIC): 3.475146, Bayes (BIC): 3.521946
# APARCH(1,1): Akaike (AIC): 3.467837, Bayes (BIC): 3.532187
# GARCH-M(1,1): Akaike (AIC): 3.477146, Bayes (BIC): 3.535646
cat("Best Model is ----> ** garch(1,1)** ")
## Best Model is ----> ** garch(1,1)**
# Significant: (AR1, AR2, MA1, MA2, omega, alpha1, beta1, shape)
garch_residuals <- residuals(fit_garch, standardize = TRUE)
#Heteroskedasticity
ArchTest(garch_residuals, lags = 10)
##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: garch_residuals
## Chi-squared = 16.549, df = 10, p-value = 0.08495
# p-value = 0.08495 > 0.05 => No presence of ARCH effects
# White noise
Box.test(garch_residuals, lag = 10, type = "Ljung-Box")
##
## Box-Ljung test
##
## data: garch_residuals
## X-squared = 11.113, df = 10, p-value = 0.3488
# p-value = 0.3488 => Residuals are white noise
# ACF
asset_name = "(Garch Residuals - Reliance) - "
acf(garch_residuals, main = paste(asset_name, "ACF"))
# no significant autocorrelation
Garch Model - Index (Nifty 50) With arma(0,0)
# Mean model spec (ARMA(0,0) with mean)
Mean_ <- list(armaOrder = c(0,0), include.mean = TRUE)
# GARCH(1,1)
Spec_garch <- ugarchspec(
variance.model = list(model = "sGARCH", garchOrder = c(1,1)),
mean.model = Mean_,
distribution.model = "std"
)
Fit_garch <- ugarchfit(Spec_garch, data = index)
show(Fit_garch)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.086971 0.027433 3.1704 0.001523
## omega 0.088488 0.052429 1.6878 0.091456
## alpha1 0.142605 0.056595 2.5197 0.011744
## beta1 0.746737 0.110965 6.7295 0.000000
## shape 6.367347 1.369398 4.6497 0.000003
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.086971 0.034971 2.48693 0.012885
## omega 0.088488 0.127845 0.69215 0.488842
## alpha1 0.142605 0.116226 1.22696 0.219838
## beta1 0.746737 0.273623 2.72907 0.006351
## shape 6.367347 1.506658 4.22614 0.000024
##
## LogLikelihood : -980.0814
##
## Information Criteria
## ------------------------------------
##
## Akaike 2.4596
## Bayes 2.4889
## Shibata 2.4596
## Hannan-Quinn 2.4709
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.097 0.1476
## Lag[2*(p+q)+(p+q)-1][2] 2.146 0.2396
## Lag[4*(p+q)+(p+q)-1][5] 3.000 0.4070
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 7.782 0.005278
## Lag[2*(p+q)+(p+q)-1][5] 9.546 0.012027
## Lag[4*(p+q)+(p+q)-1][9] 12.299 0.015499
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.5249 0.500 2.000 0.4688
## ARCH Lag[5] 1.4818 1.440 1.667 0.5971
## ARCH Lag[7] 2.7512 2.315 1.543 0.5617
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 3.2984
## Individual Statistics:
## mu 0.05591
## omega 0.84240
## alpha1 1.34712
## beta1 1.46113
## shape 1.15077
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.773 0.07653 *
## Negative Sign Bias 0.623 0.53346
## Positive Sign Bias 1.139 0.25508
## Joint Effect 6.241 0.10044
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 27.54 0.0927
## 2 30 33.87 0.2441
## 3 40 49.79 0.1155
## 4 50 52.75 0.3314
##
##
## Elapsed time : 0.07450485
# GJR-GARCH(1,1)
Spec_gjr <- ugarchspec(
variance.model = list(model = "gjrGARCH", garchOrder = c(1,1)),
mean.model = Mean_,
distribution.model = "std"
)
Fit_gjr <- ugarchfit(Spec_gjr, data = index)
show(Fit_gjr)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.076813 0.026723 2.87435 0.004049
## omega 0.112895 0.042110 2.68097 0.007341
## alpha1 0.027122 0.038618 0.70232 0.482477
## beta1 0.698295 0.092293 7.56605 0.000000
## gamma1 0.245573 0.086675 2.83325 0.004608
## shape 6.474822 1.374654 4.71014 0.000002
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.076813 0.030867 2.48852 0.012828
## omega 0.112895 0.079765 1.41534 0.156970
## alpha1 0.027122 0.055474 0.48892 0.624897
## beta1 0.698295 0.181859 3.83975 0.000123
## gamma1 0.245573 0.103317 2.37688 0.017460
## shape 6.474822 1.672171 3.87211 0.000108
##
## LogLikelihood : -973.3209
##
## Information Criteria
## ------------------------------------
##
## Akaike 2.4452
## Bayes 2.4803
## Shibata 2.4451
## Hannan-Quinn 2.4587
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.961 0.1614
## Lag[2*(p+q)+(p+q)-1][2] 1.989 0.2646
## Lag[4*(p+q)+(p+q)-1][5] 2.749 0.4547
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 11.08 0.000873
## Lag[2*(p+q)+(p+q)-1][5] 12.73 0.001793
## Lag[4*(p+q)+(p+q)-1][9] 14.62 0.004419
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.2778 0.500 2.000 0.5982
## ARCH Lag[5] 1.2406 1.440 1.667 0.6631
## ARCH Lag[7] 1.5822 2.315 1.543 0.8047
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 3.553
## Individual Statistics:
## mu 0.09859
## omega 0.98160
## alpha1 0.81374
## beta1 1.80582
## gamma1 1.43291
## shape 1.41635
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.49 1.68 2.12
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.6624 0.0079161 ***
## Negative Sign Bias 0.4702 0.6383198
## Positive Sign Bias 3.7981 0.0001569 ***
## Joint Effect 14.8883 0.0019147 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 23.14 0.2310
## 2 30 38.06 0.1209
## 3 40 38.80 0.4789
## 4 50 57.74 0.1836
##
##
## Elapsed time : 0.1982841
# EGARCH(1,1)
Spec_egarch <- ugarchspec(
variance.model = list(model = "eGARCH", garchOrder = c(1,1)),
mean.model = Mean_,
distribution.model = "std"
)
Fit_egarch <- ugarchfit(Spec_egarch, data = index)
show(Fit_egarch)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.075038 0.024583 3.0525 0.002270
## omega -0.062178 0.027737 -2.2417 0.024980
## alpha1 -0.161581 0.045522 -3.5495 0.000386
## beta1 0.852958 0.058093 14.6826 0.000000
## gamma1 0.207399 0.073081 2.8379 0.004541
## shape 6.559440 1.380537 4.7514 0.000002
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.075038 0.025597 2.9315 0.003374
## omega -0.062178 0.040929 -1.5191 0.128727
## alpha1 -0.161581 0.050755 -3.1836 0.001455
## beta1 0.852958 0.091207 9.3519 0.000000
## gamma1 0.207399 0.088467 2.3444 0.019059
## shape 6.559440 1.766986 3.7122 0.000205
##
## LogLikelihood : -970.4638
##
## Information Criteria
## ------------------------------------
##
## Akaike 2.4381
## Bayes 2.4732
## Shibata 2.4380
## Hannan-Quinn 2.4516
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.570 0.2102
## Lag[2*(p+q)+(p+q)-1][2] 1.578 0.3434
## Lag[4*(p+q)+(p+q)-1][5] 2.330 0.5428
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 13.77 0.0002070
## Lag[2*(p+q)+(p+q)-1][5] 14.81 0.0004985
## Lag[4*(p+q)+(p+q)-1][9] 16.57 0.0014751
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.01996 0.500 2.000 0.8876
## ARCH Lag[5] 0.81261 1.440 1.667 0.7892
## ARCH Lag[7] 1.04197 2.315 1.543 0.9066
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 3.9091
## Individual Statistics:
## mu 0.12360
## omega 1.31789
## alpha1 0.19156
## beta1 0.13578
## gamma1 0.05584
## shape 1.35655
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.49 1.68 2.12
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.47835 0.0134053 **
## Negative Sign Bias 0.02552 0.9796455
## Positive Sign Bias 4.43722 0.0000104 ***
## Joint Effect 19.68964 0.0001968 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 18.30 0.5024
## 2 30 35.52 0.1881
## 3 40 41.90 0.3463
## 4 50 49.50 0.4532
##
##
## Elapsed time : 0.07294011
# EGARCH(2,1)
Spec_egarch21 <- ugarchspec(
variance.model = list(model = "eGARCH", garchOrder = c(2,1)),
mean.model = Mean_,
distribution.model = "std"
)
Fit_egarch21 <- ugarchfit(Spec_egarch21, data = index)
show(Fit_egarch21)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(2,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.071834 0.025283 2.8412 0.004495
## omega -0.007678 0.002737 -2.8050 0.005032
## alpha1 -0.138911 0.053207 -2.6108 0.009034
## alpha2 0.087127 0.054871 1.5879 0.112319
## beta1 0.986583 0.004026 245.0453 0.000000
## gamma1 0.330537 0.088245 3.7457 0.000180
## gamma2 -0.276857 0.079324 -3.4902 0.000483
## shape 6.575689 1.144731 5.7443 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.071834 0.030764 2.3350 0.019541
## omega -0.007678 0.003955 -1.9413 0.052226
## alpha1 -0.138911 0.049178 -2.8246 0.004733
## alpha2 0.087127 0.054666 1.5938 0.110981
## beta1 0.986583 0.007436 132.6786 0.000000
## gamma1 0.330537 0.118069 2.7995 0.005118
## gamma2 -0.276857 0.127832 -2.1658 0.030327
## shape 6.575689 1.694131 3.8815 0.000104
##
## LogLikelihood : -965.2666
##
## Information Criteria
## ------------------------------------
##
## Akaike 2.4301
## Bayes 2.4769
## Shibata 2.4299
## Hannan-Quinn 2.4481
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.5739 0.4487
## Lag[2*(p+q)+(p+q)-1][2] 0.6042 0.6466
## Lag[4*(p+q)+(p+q)-1][5] 1.9230 0.6367
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 18.06 2.145e-05
## Lag[2*(p+q)+(p+q)-1][8] 20.63 7.902e-05
## Lag[4*(p+q)+(p+q)-1][14] 23.63 3.032e-04
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.2602 0.500 2.000 0.6100
## ARCH Lag[6] 1.8954 1.461 1.711 0.5143
## ARCH Lag[8] 4.1338 2.368 1.583 0.3560
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 2.3115
## Individual Statistics:
## mu 0.0669
## omega 0.1776
## alpha1 0.5627
## alpha2 0.4603
## beta1 0.1151
## gamma1 0.4512
## gamma2 0.5303
## shape 1.0480
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.89 2.11 2.59
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.9463 0.05197 *
## Negative Sign Bias 0.7758 0.43807
## Positive Sign Bias 2.4477 0.01459 **
## Joint Effect 6.6923 0.08238 *
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 25.79 0.13610
## 2 30 35.67 0.18348
## 3 40 42.10 0.33842
## 4 50 62.98 0.08648
##
##
## Elapsed time : 0.1181722
# EGARCH(1,2)
Spec_egarch12 <- ugarchspec(
variance.model = list(model = "eGARCH", garchOrder = c(1,2)),
mean.model = Mean_,
distribution.model = "std"
)
Fit_egarch12 <- ugarchfit(Spec_egarch12, data = index)
show(Fit_egarch12)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(1,2)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.075543 0.026195 2.8839 0.003928
## omega -0.061918 0.030188 -2.0511 0.040257
## alpha1 -0.168837 0.048819 -3.4584 0.000543
## beta1 0.514169 0.173006 2.9720 0.002959
## beta2 0.342431 0.183330 1.8678 0.061784
## gamma1 0.256316 0.083528 3.0686 0.002150
## shape 6.543926 1.380310 4.7409 0.000002
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.075543 0.029329 2.5757 0.010004
## omega -0.061918 0.044916 -1.3785 0.168039
## alpha1 -0.168837 0.051337 -3.2888 0.001006
## beta1 0.514169 0.147121 3.4949 0.000474
## beta2 0.342431 0.190151 1.8008 0.071729
## gamma1 0.256316 0.107701 2.3799 0.017317
## shape 6.543926 1.745035 3.7500 0.000177
##
## LogLikelihood : -969.2126
##
## Information Criteria
## ------------------------------------
##
## Akaike 2.4375
## Bayes 2.4784
## Shibata 2.4373
## Hannan-Quinn 2.4532
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.521 0.2174
## Lag[2*(p+q)+(p+q)-1][2] 1.535 0.3530
## Lag[4*(p+q)+(p+q)-1][5] 2.404 0.5264
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 16.07 6.114e-05
## Lag[2*(p+q)+(p+q)-1][8] 18.04 3.752e-04
## Lag[4*(p+q)+(p+q)-1][14] 20.69 1.478e-03
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.06981 0.500 2.000 0.7916
## ARCH Lag[6] 1.27562 1.461 1.711 0.6703
## ARCH Lag[8] 3.71719 2.368 1.583 0.4188
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 3.7881
## Individual Statistics:
## mu 0.09915
## omega 0.98817
## alpha1 0.34112
## beta1 0.15599
## beta2 0.18276
## gamma1 0.05367
## shape 1.35669
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.69 1.9 2.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.3619 1.842e-02 **
## Negative Sign Bias 0.3979 6.908e-01
## Positive Sign Bias 3.9482 8.572e-05 ***
## Joint Effect 15.7471 1.278e-03 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 21.35 0.31794
## 2 30 47.73 0.01569
## 3 40 42.10 0.33842
## 4 50 62.23 0.09707
##
##
## Elapsed time : 0.07548499
# IGARCH(1,1)
Spec_igarch <- ugarchspec(
variance.model = list(model = "iGARCH", garchOrder = c(1,1)),
mean.model = Mean_,
distribution.model = "std"
)
Fit_igarch <- ugarchfit(Spec_igarch, data = index)
show(Fit_igarch)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : iGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.072751 0.026934 2.7011 0.006912
## omega 0.002590 0.003066 0.8448 0.398222
## alpha1 0.035537 0.020640 1.7218 0.085114
## beta1 0.964463 NA NA NA
## shape 5.664086 1.133145 4.9986 0.000001
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.072751 0.028201 2.57977 0.009887
## omega 0.002590 0.005517 0.46947 0.638737
## alpha1 0.035537 0.039739 0.89427 0.371176
## beta1 0.964463 NA NA NA
## shape 5.664086 1.252941 4.52063 0.000006
##
## LogLikelihood : -983.4754
##
## Information Criteria
## ------------------------------------
##
## Akaike 2.4656
## Bayes 2.4890
## Shibata 2.4656
## Hannan-Quinn 2.4746
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.07165 0.7889
## Lag[2*(p+q)+(p+q)-1][2] 0.18694 0.8623
## Lag[4*(p+q)+(p+q)-1][5] 1.38387 0.7684
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 53.92 2.092e-13
## Lag[2*(p+q)+(p+q)-1][5] 54.20 3.886e-15
## Lag[4*(p+q)+(p+q)-1][9] 55.10 3.508e-14
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.09079 0.500 2.000 0.7632
## ARCH Lag[5] 0.69157 1.440 1.667 0.8260
## ARCH Lag[7] 0.82112 2.315 1.543 0.9408
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.5157
## Individual Statistics:
## mu 0.07006
## omega 0.06002
## alpha1 0.07509
## shape 0.77818
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.07 1.24 1.6
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.923 0.0547969 *
## Negative Sign Bias 2.123 0.0340671 **
## Positive Sign Bias 3.861 0.0001220 ***
## Joint Effect 20.832 0.0001141 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 19.90 0.400686
## 2 30 45.63 0.025534
## 3 40 45.69 0.213953
## 4 50 78.59 0.004629
##
##
## Elapsed time : 0.02559805
# APARCH(1,1)
Spec_aparch <- ugarchspec(
variance.model = list(model = "apARCH", garchOrder = c(1,1)),
mean.model = Mean_,
distribution.model = "std"
)
Fit_aparch <- ugarchfit(Spec_aparch, data = index)
show(Fit_aparch)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : apARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.075715 0.028060 2.6984 0.006968
## omega 0.113991 0.040094 2.8431 0.004468
## alpha1 0.113456 0.034207 3.3168 0.000911
## beta1 0.781897 0.065416 11.9527 0.000000
## gamma1 0.864226 0.239032 3.6155 0.000300
## delta 0.906783 0.264696 3.4257 0.000613
## shape 6.813144 1.470107 4.6345 0.000004
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.075715 0.033530 2.2581 0.023939
## omega 0.113991 0.054491 2.0919 0.036444
## alpha1 0.113456 0.038318 2.9609 0.003067
## beta1 0.781897 0.085786 9.1145 0.000000
## gamma1 0.864226 0.219328 3.9403 0.000081
## delta 0.906783 0.229943 3.9435 0.000080
## shape 6.813144 1.917692 3.5528 0.000381
##
## LogLikelihood : -969.1142
##
## Information Criteria
## ------------------------------------
##
## Akaike 2.4372
## Bayes 2.4782
## Shibata 2.4371
## Hannan-Quinn 2.4530
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.350 0.2453
## Lag[2*(p+q)+(p+q)-1][2] 1.362 0.3945
## Lag[4*(p+q)+(p+q)-1][5] 2.163 0.5805
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 18.67 1.554e-05
## Lag[2*(p+q)+(p+q)-1][5] 19.55 2.577e-05
## Lag[4*(p+q)+(p+q)-1][9] 21.33 9.115e-05
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.006054 0.500 2.000 0.9380
## ARCH Lag[5] 0.905644 1.440 1.667 0.7611
## ARCH Lag[7] 1.145379 2.315 1.543 0.8888
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 3.472
## Individual Statistics:
## mu 0.1995
## omega 1.0528
## alpha1 1.4247
## beta1 1.4201
## gamma1 0.1115
## delta 1.1990
## shape 1.5458
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.69 1.9 2.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.1535 3.158e-02 **
## Negative Sign Bias 0.3459 7.295e-01
## Positive Sign Bias 4.5462 6.311e-06 ***
## Joint Effect 20.8738 1.118e-04 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 19.10 0.4504
## 2 30 35.44 0.1904
## 3 40 38.90 0.4744
## 4 50 49.25 0.4631
##
##
## Elapsed time : 0.2696431
# APARCH(2,1)
Spec_aparch21 <- ugarchspec(
variance.model = list(model = "apARCH", garchOrder = c(2,1)),
mean.model = Mean_,
distribution.model = "std"
)
Fit_aparch21 <- ugarchfit(Spec_aparch21, data = index)
show(Fit_aparch21)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : apARCH(2,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.07577 0.027633 2.741980 0.006107
## omega 0.11300 0.039653 2.849740 0.004376
## alpha1 0.11187 0.034184 3.272491 0.001066
## alpha2 0.00000 0.000465 0.000000 1.000000
## beta1 0.78374 0.065136 12.032462 0.000000
## gamma1 0.88236 0.246097 3.585420 0.000337
## gamma2 0.44830 38.034031 0.011787 0.990596
## delta 0.91588 0.259913 3.523797 0.000425
## shape 6.80708 1.465358 4.645339 0.000003
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.07577 0.032578 2.325796 0.020029
## omega 0.11300 0.053933 2.095213 0.036152
## alpha1 0.11187 0.038306 2.920366 0.003496
## alpha2 0.00000 0.000004 0.000036 0.999971
## beta1 0.78374 0.085290 9.189202 0.000000
## gamma1 0.88236 0.226288 3.899271 0.000096
## gamma2 0.44830 44.257006 0.010130 0.991918
## delta 0.91588 0.226492 4.043778 0.000053
## shape 6.80708 1.910103 3.563726 0.000366
##
## LogLikelihood : -969.0588
##
## Information Criteria
## ------------------------------------
##
## Akaike 2.4421
## Bayes 2.4947
## Shibata 2.4418
## Hannan-Quinn 2.4623
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.353 0.2448
## Lag[2*(p+q)+(p+q)-1][2] 1.364 0.3938
## Lag[4*(p+q)+(p+q)-1][5] 2.163 0.5805
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 18.57 0.0000164
## Lag[2*(p+q)+(p+q)-1][8] 20.58 0.0000814
## Lag[4*(p+q)+(p+q)-1][14] 23.24 0.0003758
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.1096 0.500 2.000 0.7406
## ARCH Lag[6] 1.2219 1.461 1.711 0.6851
## ARCH Lag[8] 3.8538 2.368 1.583 0.3974
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 3.7708
## Individual Statistics:
## mu 0.2034
## omega 1.0357
## alpha1 1.4254
## alpha2 1.2807
## beta1 1.4123
## gamma1 0.0924
## gamma2 0.1291
## delta 1.2204
## shape 1.5466
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.1 2.32 2.82
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.1766 2.980e-02 **
## Negative Sign Bias 0.3414 7.329e-01
## Positive Sign Bias 4.6029 4.848e-06 ***
## Joint Effect 21.3982 8.702e-05 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 18.60 0.4827
## 2 30 37.01 0.1458
## 3 40 37.50 0.5383
## 4 50 50.62 0.4093
##
##
## Elapsed time : 0.4533129
# APARCH(1,2)
Spec_aparch12 <- ugarchspec(
variance.model = list(model = "apARCH", garchOrder = c(1,2)),
mean.model = Mean_,
distribution.model = "std"
)
Fit_aparch12 <- ugarchfit(Spec_aparch12, data = index)
show(Fit_aparch12)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : apARCH(1,2)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.076573 0.026434 2.8968 0.003770
## omega 0.118800 0.042991 2.7633 0.005721
## alpha1 0.127453 0.038485 3.3117 0.000927
## beta1 0.494559 0.260792 1.8964 0.057911
## beta2 0.268398 0.246165 1.0903 0.275574
## gamma1 0.860149 0.251411 3.4213 0.000623
## delta 0.942728 0.259730 3.6296 0.000284
## shape 6.760350 1.442128 4.6878 0.000003
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.076573 0.029746 2.57423 0.010046
## omega 0.118800 0.055064 2.15750 0.030967
## alpha1 0.127453 0.040418 3.15334 0.001614
## beta1 0.494559 0.270765 1.82653 0.067771
## beta2 0.268398 0.274658 0.97721 0.328467
## gamma1 0.860149 0.237725 3.61825 0.000297
## delta 0.942728 0.240239 3.92412 0.000087
## shape 6.760350 1.909525 3.54033 0.000400
##
## LogLikelihood : -968.5013
##
## Information Criteria
## ------------------------------------
##
## Akaike 2.4382
## Bayes 2.4850
## Shibata 2.4380
## Hannan-Quinn 2.4562
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.228 0.2677
## Lag[2*(p+q)+(p+q)-1][2] 1.242 0.4260
## Lag[4*(p+q)+(p+q)-1][5] 2.126 0.5890
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 20.30 6.634e-06
## Lag[2*(p+q)+(p+q)-1][8] 22.03 3.360e-05
## Lag[4*(p+q)+(p+q)-1][14] 24.49 1.886e-04
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.05684 0.500 2.000 0.8116
## ARCH Lag[6] 1.17233 1.461 1.711 0.6989
## ARCH Lag[8] 3.52349 2.368 1.583 0.4505
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 3.6626
## Individual Statistics:
## mu 0.19046
## omega 0.89200
## alpha1 1.38743
## beta1 1.25473
## beta2 1.27688
## gamma1 0.03721
## delta 1.18499
## shape 1.56919
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.89 2.11 2.59
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.18044 2.952e-02 **
## Negative Sign Bias 0.02532 9.798e-01
## Positive Sign Bias 4.53691 6.588e-06 ***
## Joint Effect 20.80481 1.156e-04 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 22.15 0.2770
## 2 30 38.74 0.1068
## 3 40 44.19 0.2615
## 4 50 46.00 0.5953
##
##
## Elapsed time : 0.2729409
# APARCH(2,2)
Spec_aparch22 <- ugarchspec(
variance.model = list(model = "apARCH", garchOrder = c(2,2)),
mean.model = Mean_,
distribution.model = "std"
)
Fit_aparch22 <- ugarchfit(Spec_aparch22, data = index)
show(Fit_aparch22)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : apARCH(2,2)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.076573 0.026438 2.8963 0.003776
## omega 0.118820 0.048438 2.4530 0.014167
## alpha1 0.127437 0.037281 3.4182 0.000630
## alpha2 0.000000 0.032966 0.0000 1.000000
## beta1 0.494533 0.136703 3.6176 0.000297
## beta2 0.268405 0.149137 1.7997 0.071905
## gamma1 0.860248 0.250584 3.4330 0.000597
## gamma2 0.985258 0.000263 3742.5572 0.000000
## delta 0.942816 0.260923 3.6134 0.000302
## shape 6.760991 1.444956 4.6790 0.000003
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.076573 0.029631 2.5842 0.009761
## omega 0.118820 0.070414 1.6874 0.091517
## alpha1 0.127437 0.034923 3.6491 0.000263
## alpha2 0.000000 0.033881 0.0000 1.000000
## beta1 0.494533 0.049752 9.9399 0.000000
## beta2 0.268405 0.148956 1.8019 0.071559
## gamma1 0.860248 0.234568 3.6674 0.000245
## gamma2 0.985258 0.000983 1002.6990 0.000000
## delta 0.942816 0.250230 3.7678 0.000165
## shape 6.760991 1.901407 3.5558 0.000377
##
## LogLikelihood : -968.5013
##
## Information Criteria
## ------------------------------------
##
## Akaike 2.4432
## Bayes 2.5017
## Shibata 2.4429
## Hannan-Quinn 2.4657
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.228 0.2678
## Lag[2*(p+q)+(p+q)-1][2] 1.241 0.4261
## Lag[4*(p+q)+(p+q)-1][5] 2.125 0.5891
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 20.29 6.639e-06
## Lag[2*(p+q)+(p+q)-1][11] 23.52 7.340e-05
## Lag[4*(p+q)+(p+q)-1][19] 25.53 9.291e-04
## d.o.f=4
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[5] 1.338 0.500 2.000 0.2474
## ARCH Lag[7] 1.466 1.473 1.746 0.6315
## ARCH Lag[9] 4.389 2.402 1.619 0.3424
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 4.4176
## Individual Statistics:
## mu 0.19042
## omega 0.89206
## alpha1 1.38779
## alpha2 1.24923
## beta1 1.25489
## beta2 1.27703
## gamma1 0.03716
## gamma2 0.74938
## delta 1.18514
## shape 1.56857
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.29 2.54 3.05
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.18074 2.949e-02 **
## Negative Sign Bias 0.02541 9.797e-01
## Positive Sign Bias 4.53743 6.573e-06 ***
## Joint Effect 20.80943 1.153e-04 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 22.15 0.2770
## 2 30 38.74 0.1068
## 3 40 44.19 0.2615
## 4 50 46.00 0.5953
##
##
## Elapsed time : 0.496716
# GARCH-in-Mean(1,1)
Mean_garchm <- list(armaOrder = c(0,0), include.mean = TRUE, archm = TRUE)
Spec_garchm <- ugarchspec(
variance.model = list(model = "sGARCH", garchOrder = c(1,1)),
mean.model = Mean_garchm,
distribution.model = "std"
)
Fit_garchm <- ugarchfit(Spec_garchm, data = index)
show(Fit_garchm)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu -0.055388 0.15720 -0.35234 0.724587
## archm 0.174215 0.18886 0.92245 0.356294
## omega 0.077542 0.04842 1.60145 0.109278
## alpha1 0.128671 0.05499 2.33989 0.019289
## beta1 0.773108 0.10609 7.28705 0.000000
## shape 6.412139 1.37116 4.67643 0.000003
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu -0.055388 0.16173 -0.34246 0.732006
## archm 0.174215 0.17805 0.97846 0.327847
## omega 0.077542 0.11136 0.69633 0.486220
## alpha1 0.128671 0.10741 1.19791 0.230953
## beta1 0.773108 0.24485 3.15754 0.001591
## shape 6.412139 1.51680 4.22740 0.000024
##
## LogLikelihood : -979.6332
##
## Information Criteria
## ------------------------------------
##
## Akaike 2.4610
## Bayes 2.4961
## Shibata 2.4609
## Hannan-Quinn 2.4745
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.268 0.1321
## Lag[2*(p+q)+(p+q)-1][2] 2.370 0.2081
## Lag[4*(p+q)+(p+q)-1][5] 3.332 0.3497
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 9.707 0.001835
## Lag[2*(p+q)+(p+q)-1][5] 11.437 0.003911
## Lag[4*(p+q)+(p+q)-1][9] 13.942 0.006413
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.5092 0.500 2.000 0.4755
## ARCH Lag[5] 1.3678 1.440 1.667 0.6278
## ARCH Lag[7] 2.5910 2.315 1.543 0.5939
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 3.4809
## Individual Statistics:
## mu 0.07533
## archm 0.07329
## omega 0.75112
## alpha1 1.28240
## beta1 1.35329
## shape 1.20929
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.49 1.68 2.12
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.8363 0.06669 *
## Negative Sign Bias 0.7152 0.47470
## Positive Sign Bias 1.4797 0.13936
## Joint Effect 6.7111 0.08170 *
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 34.78 0.01484
## 2 30 43.38 0.04196
## 3 40 48.39 0.14408
## 4 50 58.74 0.16072
##
##
## Elapsed time : 0.122926
# Compare Information Criteria
infocriteria(Fit_garch)
##
## Akaike 2.459629
## Bayes 2.488879
## Shibata 2.459552
## Hannan-Quinn 2.470865
infocriteria(Fit_gjr)
##
## Akaike 2.445246
## Bayes 2.480346
## Shibata 2.445134
## Hannan-Quinn 2.458729
infocriteria(Fit_egarch)
##
## Akaike 2.438112
## Bayes 2.473212
## Shibata 2.438001
## Hannan-Quinn 2.451595
infocriteria(Fit_egarch21)
##
## Akaike 2.430129
## Bayes 2.476929
## Shibata 2.429932
## Hannan-Quinn 2.448106
infocriteria(Fit_egarch12)
##
## Akaike 2.437485
## Bayes 2.478435
## Shibata 2.437334
## Hannan-Quinn 2.453215
infocriteria(Fit_igarch)
##
## Akaike 2.465607
## Bayes 2.489007
## Shibata 2.465557
## Hannan-Quinn 2.474595
infocriteria(Fit_aparch)
##
## Akaike 2.437239
## Bayes 2.478189
## Shibata 2.437088
## Hannan-Quinn 2.452969
infocriteria(Fit_aparch21)
##
## Akaike 2.442094
## Bayes 2.494745
## Shibata 2.441846
## Hannan-Quinn 2.462319
infocriteria(Fit_aparch12)
##
## Akaike 2.438205
## Bayes 2.485006
## Shibata 2.438009
## Hannan-Quinn 2.456183
infocriteria(Fit_aparch22)
##
## Akaike 2.443199
## Bayes 2.501699
## Shibata 2.442893
## Hannan-Quinn 2.465671
infocriteria(Fit_garchm)
##
## Akaike 2.461007
## Bayes 2.496107
## Shibata 2.460896
## Hannan-Quinn 2.474490
Best Garch Model - Index (Nifty 50) with arma(0,0)
# GARCH(1,1): Akaike (AIC): 2.459629, Bayes (BIC): 2.488879
# GJR-GARCH(1,1): Akaike (AIC): 2.445246, Bayes (BIC): 2.480346
# EGARCH(1,1): Akaike (AIC): 2.438112, Bayes (BIC): 2.473212
# EGARCH(2,1): Akaike (AIC): 2.438112, Bayes (BIC): 2.473212
# EGARCH(1,2): Akaike (AIC): 2.438112, Bayes (BIC): 2.473212
# IGARCH(1,1): Akaike (AIC): 2.465607, Bayes (BIC): 2.489007
# APARCH(1,1): Akaike (AIC): 2.437239, Bayes (BIC): 2.478189
# APARCH(2,1): Akaike (AIC): 2.442094, Bayes (BIC): 2.494745
# APARCH(1,2): Akaike (AIC): 2.438205, Bayes (BIC): 2.485006
# APARCH(2,2): Akaike (AIC): 2.443199, Bayes (BIC): 2.501699
# GARCH-M(1,1): Akaike (AIC): 2.461007, Bayes (BIC): 2.496107
cat("Best Model is ----> ** aparch(1,1)** ")
## Best Model is ----> ** aparch(1,1)**
# aparch (1,1) has lowest AIC, then egarch(1,1).
# also, apARCH has all significant coefficients while eGARCH's ω (omega) is not significant.
# Tried Aparch(1,2), Aparch(2,1) - they have same AIC value and same case with egarach(1,2), egarach(2,1)
Garch_residuals <- residuals(Fit_aparch, standardize = TRUE)
#Heteroskedasticity
ArchTest(Garch_residuals, lags = 10)
##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: Garch_residuals
## Chi-squared = 29.662, df = 10, p-value = 0.0009728
# p-value = 0.0009728 < 0.05 => presence of ARCH effects
# aparch(1,1) and egarch(1,1) are the two best model, but both have arch effect, even their high order.
# White noise
Box.test(Garch_residuals, lag = 10, type = "Ljung-Box")
##
## Box-Ljung test
##
## data: Garch_residuals
## X-squared = 4.2985, df = 10, p-value = 0.9329
# p-value = 0.9329 => Residuals are white noise
# ACF
asset_name = "(Garch Residuals - Nifty 50) - "
acf(Garch_residuals, main = paste(asset_name, "ACF"))
# no significant autocorrelation
